Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24);(1,2,7,8)(3,6,5,4)(9,11)(10,13)(12,16)(14,15);(1,3,7,5)(2,4,8,6)(10,14)(13,15)(17,18,20,22)(19,21,24,23);(1,4,7,6)(2,3,8,5)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24);(1,5,7,3)(2,4,8,6)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,24)(22,23);(1,6,7,4)(2,3,8,5)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72238573, #72238574, #72238592 |
| maximal normal subgroup of | #72238573, #72238574, #72238592 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7756828);